The squares seem to be plaster?

As above, sure it's not tile?

If it is plaster, the easiest fix is to skim coat with drywall compound to fill the gaps and paint.

If tile, scrape or sand off paint till you have a uniform surface, then repaint with a product designed to adhere to tile, such as Benjamin Moore bonding primer and top with the finish paint.

Is there a way I can check if it’s tile vs plaster without ruining a piece (if plaster)?

Here’s a few more shots

My FIL's house, built in the 60s had pseudo-tile wall board in the bathroom. In a picture, it would look like tile. If you actually saw it, you could tell, particularly since the board was decaying around the faucet. But, the grout lines were indented on the board.

If it's wallboard, there are probably seams. If the installation was well-done, the seams might be hard to find. If the bathroom is tiny, perhaps no wall took more than one sheet of wall board.

In addition, plaster will feel warmer to the touch and have a different sound when tapped with a wood stick.

Oh this is so helpful, thank you Sigrid! I will look into this.

My neighbor's house, built in 1940 had a tile pattern wainscot inscribed in the tile in the kitchen and bathroom around the sink and toilet areas The bathtub had some real tile around it, that did not look quite the same. There was no shower in that house.